From now on we will refer to [REF ] as “plain” FR curvature, as the generality of Forman's construction actually allows for a more general definition [1]}. This generalisation, usually called “augmented” curvature, adds terms that account for the cycles of the network. A network is originally a one-dimensional object (it has 0-dimensional nodes and 1-dimensional edges), and we may turn it into a two-dimensional one by inserting flat faces that fill the spaces encircled by its cycles. This new two-dimensional object is too a CW-complex, and Forman's work allows us to compute the curvature of its edges. Now, if we assume that this augmented network is still quasiconvex (that is, any two inserted faces share at most one edge), applying Forman's quasiconvex formula gives the following augmented FR curvature for an edge \(e=(v_1,v_2)\) :
